Pólya s Random Walk Theorem Jonathan ovak Abstract. This note presents a proof of Pólya s random walk theorem using classical methods from special function theory and asymptotic analysis. 1. ITRODUCTIO. This note is about a remarkable law of nature discovered by George Pólya [6]. Consider a particle situated at a given point of the integer lattice Z d. Suppose that, at each tick of the clock, the particle jumps to a randomly selected neighboring lattice point, with equal probability of jumping in any direction. In other words, this particle is executing the simple random walk on Z d. A random walk is said to be recurrent if it returns to its initial position with probability one. A random walk that is not recurrent is called transient. Pólya s classic result [6] is the following. Theorem 1. The simple random walk on Z d is recurrent in dimensions d = 1, 2 and transient in dimension d 3. Pólya s theorem is a foundational result in the theory of random walks, and many proofs are available. This note presents a new proof of Pólya s theorem using techniques developed by de Moivre and Laplace in the 18th and 19th centuries in order to to establish the basic limit theorems of probability theory; see [5, Chapter 2]. These classical methods have returned to the forefront of contemporary probability, where new universality classes of limit theorems are being investigated via the asymptotic analysis of exact formulas [4]. Thus, in a sense, the proof of Pólya s theorem given here is both more classical and more modern than the arguments one finds in textbooks. 2. LOOP DECOMPOSITIO. Let E denote the event that the simple random walk on Z d returns to its initial position, and put p = Prob(E). Forn 1, let E n be the event that the random walk returns to its initial position for the first time after n steps. It is convenient to set E =, corresponding to the fact that the initial position of the random walk does not count as a return (if it did, the return probability of any random walk would be one). The events E n are mutually exclusive for different values of n, and E = n E n. Hence, p = n p n, where p n = Prob(E n ). http://dx.doi.org/1.4169/amer.math.monthly.121.8.711 MSC: Primary 6G5 October 214] OTES 711
A loop on Z d is a walk that begins and ends at a given point. It is convenient to consider walks of length zero as loops; such loops are called trivial. A nontrivial loop is indecomposable if it is not the concatenation of two nontrivial loops. Choose a particular point of Z d, and let l n denote the number of loops of length n based at this point. Let r n denote the number of these that are indecomposable. ote that l = 1 while r =. Since any nontrivial loop is the concatenation of an indecomposable loop followed by a (possibly trivial) loop, the counts l n and r n are related by l n = n r k l n k k= for all n 1. Dividing both sides of this equation by (2d) n, the total number of length n walks emanating from a given point of Z d, we obtain the relation q n = n p k q n k k= for all n 1, where, as above, p n is the probability that the random walk returns to its initial position for the first time after n steps, while q n is the probability that the random walk is located at its original position after n steps. We introduce the generating functions P(z) = p n z n and Q(z) = n= q n z n. The relation between p n and q n is then equivalent to the identity P(z)Q(z) = Q(z) 1 in the algebra Q[[z]] of formal power series. Since p n q n 1, each of these series has radius of convergence at least one, and the above may be considered as an identity in the algebra of analytic functions on the open unit disk in C. The function Q(z) is nonvanishing for z in the interval [, 1), and hence, we have Since n= P(z) = 1 1, z [, 1). Q(z) P(1) = p n = p, n= Abel s power series theorem applies, and we have 1 p = lim P(z) = 1 z 1 lim Q(z). z [,1) z 1 z [,1) The limit in the denominator is either + or a positive real number. In the former case we have p = 1 (recurrence) and in the latter p < 1 (transience). 712 c THE MATHEMATICAL ASSOCIATIO OF AMERICA [Monthly 121
3. EXPOETIAL LOOP GEERATIG FUCTIO. In order to analyze the limit in question, we need a tractable representation of the function Q(z). This amounts to finding an expression for the loop generating function L(z) = l n z n. n= Indeed, Q(z) = L( z 2d ). While the ordinary generating function L(z) is difficult to analyze directly, the exponential loop generating function E(z) = n= l n z n n! is quite accessible. This is because any loop on Z d is a shuffle of loops on Z 1, and products of exponential generating functions correspond to shuffles. This is a basic property of exponential generating functions that we will review in the specific case at hand. For a general treatment, the reader is referred to [7, Chapter 5]. In this paragraph, it is important to make the dependence on d explicit, so we write l (d) n for the number of length n loops on Z d and E d (z) for the exponential generating function of this sequence. Let us consider the case d = 2. A loop on Z 2 is a closed walk that takes unit steps in two directions, horizontal and vertical. A length n loop on Z 2 is made up of some number k of horizontal steps together with n k vertical steps. The k horizontal steps constitute a length k loop on Z, and the n k vertical steps constitute a length n k loop on Z. Thus, the number of length n loops on Z 2 that take k horizontal and n k vertical steps is ( ) n k l (1) k l(1) n k since specifying the times at which the k horizontal steps occur uniquely determines the times at which the n k vertical steps occur. The total number of length n loops on Z 2 is therefore l (2) n = n k= ( ) n l (1) k k l(1) n k. This is equivalent to the generating function identity E 2 (z) = E 1 (z) 2. The same reasoning applies for any d, and in general we have E d (z) = E 1 (z) d. Counting loops in one dimension is easy, l (1) n = {( 2k ) k, if n = 2k is even,, if n is odd. October 214] OTES 713
Indeed, any loop on Z consists of k positive steps and k negative steps for some k, and the times at which the positive steps occur determine the times at which the negative steps occur. Thus, E 1 (z) = k= ( ) 2k z 2k k (2k)! = z 2k k!k!. k= ow a minor miracle occurs: The exponential generating function for lattice walks in one dimension is a modified Bessel function of the first kind. The modified Bessel function of the first kind, usually denoted I α (z), is one of two linearly independent solutions to the second-order differential equation (z 2 d2 dz + z d ) 2 dz (z2 + α 2 ) F(z) =, α C. This differential equation is known as the modified Bessel equation; it appears in a multitude of physical problems and was exhaustively studied by 19th century mathematicians. An excellent reference on this subject is [1, Chapter 4]. It is known that the modified Bessel function admits both a series representation, I α (z) = k= ( z 2 )2k+α k!ɣ(k + α + 1), and an integral representation, I α (z) = ( z 2 )α πɣ(α + 1 ) e (cos θ)z (sin θ) 2α dθ. 2 From the series representation, we see that E 1 (z) = I (2z), and hence, E(z) = I (2z) d. 4. BOREL TRASFORM. We now have a representation of the exponential generating function E(z) counting loops on Z d in terms of a standard mathematical object, the modified Bessel function I (z). What we need, however, is a representation of the ordinary loop generating function L(z). The integral transform (B f )(z) = f (tz)e t dt, which looks like the Laplace transform of f but with the z-parameter in the wrong place, converts exponential generating functions into ordinary generating functions. To see why, write out the Maclaurin series of f (tz), interchanging integration and summation, to obtain (B f )(z) = n= f (n) () zn n! t n e t dt, 714 c THE MATHEMATICAL ASSOCIATIO OF AMERICA [Monthly 121
and use the fact that t n e t dt = n!. The transform f B f was invented by Borel in order to sum divergent series [3, p. 55]. In our case, the Borel transform produces the formula L(z) = BE(z) = BI (2z) d = which in turn leads to the integral representation Q(z) = L I (2tz) d e t dt, ( z ) ( ) tz d = I e t dt. 2d d 5. THE LAPLACE PRICIPLE. We will now use the integral representation just obtained to determine whether the limit under consideration is finite or infinite. It suffices to answer this question for the tail integral ( ) tz d I e t dt,. d For large, the behavior of the tail integral is in turn determined by the behavior of the integrand as t. In order to estimate the integrand, we invoke the formula ( ) tz I = 1 e tf(θ) dθ, d π where f (θ) = z cos θ, and estimate this integral as t using a basic technique of d asymptotic analysis known as Laplace s method. The function f (θ) is strictly maximized over the interval [,π] at the left endpoint θ =. Thus, the integrand e tf(θ) is exponentially larger at θ = than at any other point of this interval. As t, this effect becomes increasingly exaggerated, so much so that the integral localizes at θ = a in the t limit. To quantify this, note that f () =, f () <, and consider the quadratic Taylor approximation of f (θ): f (θ) f () f () θ 2 2. Replacing f (θ) with its quadratic approximation, we obtain the integral approximation e tf(θ) dθ e tf() e t f () θ2 2 dθ. Extending the integral on the right over the positive reals and ignoring the rapidly decaying error incurred results in a half a Gaussian integral, which can be computed exactly: + e t f () θ2 π 2 dθ = 2t f (). October 214] OTES 715
Thus, we expect that e tf(θ) dθ e tf() π 2t f () is an approximation of our integral whose accuracy increases as t. Laplace s principle (see, e.g., [2, 5.2]) is the statement that this is indeed the case; we have e tf(θ) dθ e tf() π 2t f (), t, where the notation F(t) G(t), t means that lim t F(t) G(t) = 1. Putting everything together, we have the asymptotic formula ( ) tz d I e t constant e t(z 1) (tz) d 2, t. d Applying the monotone convergence theorem, we find lim z 1 z [,1) e t(z 1) (tz) d 2 dt = lim z 1 z [,1) e t(z 1) (tz) d 2 dt = t d 2 dt, and thus conclude that the recurrence or transience of the simple random walk on Z d is equivalent to the divergence or convergence of the integral t d/2 dt,. Since this integral diverges for d = 1, 2 and converges for d 3, Pólya s theorem is proved. REFERECES 1. G. Andrews, R. Askey, R. Roy, Special Functions. Cambridge Univ. Press, 1999. 2.. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals. Holt, Rinehart and Winston, ew York, 1975. 3. E. Borel, Mémoire sur les series divergentes, Annales Scientifique de l É..S., 3 e série, tome 16 (1899) 131. 4. A. Borodin, V. Gorin, Lectures on integrable probability; http://arxiv.org/abs/1212.3351. 5. H. Fisher, A History of the Central Limit Theorem from Classical to Modern Probability. Springer, ew York, 211. 6. G. Pólya, Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Strassennetz, Math. Ann. 84 (1921) 149 16. 7. R. P. Stanley, Enumerative Combinatorics. Vol. 2, Cambridge Univ. Press, 1999. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 2139 jnovak@math.mit.edu 716 c THE MATHEMATICAL ASSOCIATIO OF AMERICA [Monthly 121